Nonlinear Tomography Research Articles

An efficient nonlinear Fresnel volume tomography using an improved scattering-integral algorithm

A new nonlinear Fresnel volume tomography (FVT) method is proposed in which a gradient-guided approach is used to replace the conventional equation-solving method. An improved scattering-integral algorithm is used to compute the objective function gradient with respect to the model parameters and the diagonal Hessian. The key principle of this algorithm is the transformation of the gradient computation into an accumulation of multiple vector-scalar products, where each vector is the corresponding individual Fresnel volume kernel of a source-receiver pair. This implementation has several advantages over conventional equation-solving approaches. Firstly, neither the Fresnel volumes nor the Hessian matrix must be stored in memory in advance, thereby ensuring low memory requirements. Secondly, no large equations must be iteratively solved using a general inverse algorithm such as least-squares QR decomposition (LSQR), thereby ensuring a highly efficient inversion procedure. Thirdly, the steepest-descent direction is used to ensure stable convergence. Finally, and most importantly, the preconditioning of the steepest-descent direction, which is equivalent to achieving illumination compensation, can be conveniently performed. This proposed nonlinear FVT method is applied to both synthetic and field data to invert finite-frequency traveltimes of first-arrival waveforms for the near-surface velocity structure. Shallow accuracies comparable to or better than those obtained using the equation-solving FVT or ray-based tomography are achieved. Furthermore, both the memory requirements and computation time are reduced and deeper inversion results are obtained using the diagonal Hessian.

Optimal selection of spectral lines for multispectral absorption tomography

Multispectral absorption tomography (MAT) is evolving into a mature imaging technique for flow diagnostics due to recent progress in both laser sources and nonlinear tomography. For the absorption-based technique, how to determine and utilize the most informative spectral lines are critical for successful implementation of the technique. In this work, we propose a method to select the optimal combination of spectral lines from the candidate set for MAT. We select the Gram determinant of selected spectral lines in given temperature interval as the cost function of the optimization problem, which can then be maximized by enumerating all the possible combinations of the candidate spectral lines. The numerical studies performed in this work verified the effectiveness of the proposed method, whose purpose was to achieve the best performance for the reconstruction of temperature distribution.

Transdimensional Electrical Resistivity Tomography

AbstractThis paper shows that imaging the interior of solid bodies with fully nonlinear physics can be highly beneficial compared to imaging with the equivalent linearized tomographic methods and that this is true for a variety of different types of physics. Including full nonlinearity provides interpretable uncertainties and far greater depth of image penetration into unknown targets such as the Earth's subsurface. We use an adaptively parameterized Monte Carlo method to invert electrical resistivity data for the conductivity structure of the Earth and demonstrate the method on two field data sets. Key results include the observation of directly interpretable uncertainty loops which define possible geometrical variations in the edges of isolated anomalies, hence quantifying the spatial resolution of these boundaries. These topologies of uncertainties are similar to those observed when performing fully nonlinear seismic traveltime tomography. This shows that loop‐like uncertainty topologies are expected in the solutions to a wide variety of tomographic problems, using a variety of data types and hence laws of physics (here the Laplace equation; in previous work the Eikonal or ray equations). We also show that the depth to which we can construct a tomographic image using electrical data is extended by up to a factor of 8 using nonlinear methods compared to linearized inversion using common standard linearized programs. These advantages come at the cost of significantly increased computation. All of these results are illustrated on both synthetic and real data examples.

Reconstruction for limited-data nonlinear tomographic absorption spectroscopy via deep learning

Nonlinear tomographic absorption spectroscopy (NTAS) is an emerging gas sensing technique for reactive flows that has been proven to be capable of simultaneously imaging temperature and concentration of absorbing gas. However, the nonlinear tomographic problems are typically solved with an optimization algorithm such as simulated annealing which suffers from high computational cost. This problem becomes more severe when thousands of tomographic data needs to be processed for the temporal resolution of turbulent flames. To overcome this limitation, in this work we propose a reconstruction method based on convolutional neural networks (CNN) which can take full advantage of the large amount tomographic data to build an efficient neural networks to rapidly predict the reconstruction by feeding the sinograms to it. Simulative studies were performed to investigate how the parameters will affect the performance of neural networks. The results show that CNN can effectively reduce the computational cost and at the same time achieve a similar accuracy level as SA. The successful demonstration CNN in this work indicates possible applications of other sophisticated deep neural networks such as deep belief networks (DBN) and generative adversarial networks (GAN) to nonlinear tomography. © 2018 Elsevier Ltd.

Nonlinear Acoustic Profiling of Liquid Flows

A means of nonlinear acoustic Doppler tomography is described and the results from its simulation are analyzed. The technique can be applied to the gas inclusions normally present in liquids. The distribution of the flow velocity can be reconstructed by making certain assumptions about the distribution of gas inclusions.

Imaging challenges and processing solutions in the Brazilian Equatorial Margin: A case study in the Barreirinhas Basin

The Barreirinhas Basin is located in northeast Brazil and is part of the Brazilian Equatorial Margin, a new exploration frontier with complex geology. This basin is characterized by a rugose water bottom, a fast carbonate platform, shallow gas pockets, and a complex channel network. All of these elements represent a significant challenge for velocity model building and imaging of the depositional system. From preprocessing to final imaging, high-end technologies were required to meet the processing objectives. The 3D designature and 3D deghosting were crucial to remove bubble energy and ghosts related to canyon diffractions. The velocity model building exploited offset-dependent dip information in the nonlinear slope tomography — i.e., dip-constrained tomography (DCT) — to deal with small-scale lateral velocity variations. The full-waveform inversion, up to 20 Hz, was able to efficiently capture small velocity anomalies and resolve high-spatial-resolution variations that DCT could not totally solve. Even with a detailed velocity model, some dim zones and amplitude variations were still observable in the depth-migrated image. Gas pockets, responsible for the absorption and phase distortion of the seismic signal (commonly denoted by the quality factor of attenuation, Q), were detected and delineated using volumetric Q tomography. The resultant interval Q model was consistent with the geology, and its use was beneficial in a Q-compensating Kirchhoff prestack depth migration.

Elastic-net regularization for nonlinear electrical impedance tomography with a splitting approach

ABSTRACTImage reconstruction of EIT mathematically is a typical nonlinear and severely ill-posed inverse problem. Appropriate priors or penalties are required to enable the reconstruction. The commonly used -norm can enforce the stability to preserve local smoothness, and the current -norm can enforce the sparsity to preserve sharp edges. Considering the fact that -norm penalty always makes the solution overly smooth and -norm penalty always makes the solution too sparse, elastic-net regularization approach with a convex combination term of -norm and -norm emerges for fully nonlinear EIT inverse problems. Our aim is to combine the strength of both terms: sparsity in the transform domain and smoothness in the physical domain, in an attempt to improve the reconstruction resolution and robustness to noise. Nonlinearity and non-smoothness of the generated composite minimization problem make it challenging to find an efficient numerical solution. Then we develop one simple and fast numerical optimization scheme based on the split Bregman technique for the joint penalties regularization. The method is validated using simulated data for some typical conductivity distributions. Results indicate that the proposed inversion model with an appropriate parameter choice achieves an efficient regularization solution and enhances the quality of the reconstructed image.

A parallel competitive Particle Swarm Optimization for non-linear first arrival traveltime tomography and uncertainty quantification

Seismic traveltime tomography is an optimization problem that requires large computational efforts. Therefore, linearized techniques are commonly used for their low computational cost. These local optimization methods are likely to get trapped in a local minimum as they critically depend on the initial model. On the other hand, global optimization methods based on MCMC are insensitive to the initial model but turn out to be computationally expensive. Particle Swarm Optimization (PSO) is a rather new global optimization approach with few tuning parameters that has shown excellent convergence rates and is straightforwardly parallelizable, allowing a good distribution of the workload. However, while it can traverse several local minima of the evaluated misfit function, classical implementation of PSO can get trapped in local minima at later iterations as particles inertia dim. We propose a Competitive PSO (CPSO) to help particles to escape from local minima with a simple implementation that improves swarm's diversity. The model space can be sampled by running the optimizer multiple times and by keeping all the models explored by the swarms in the different runs. A traveltime tomography algorithm based on CPSO is successfully applied on a real 3D data set in the context of induced seismicity.

On the regularization for nonlinear tomographic absorption spectroscopy

Tomographic absorption spectroscopy (TAS) has attracted increased research efforts recently due to the development in both hardware and new imaging concepts such as nonlinear tomography and compressed sensing. Nonlinear TAS is one of the emerging modality that bases on the concept of nonlinear tomography and has been successfully demonstrated both numerically and experimentally. However, all the previous demonstrations were realized using only two orthogonal projections simply for ease of implementation. In this work, we examine the performance of nonlinear TAS using other beam arrangements and test the effectiveness of the beam optimization technique that has been developed for linear TAS. In addition, so far only smoothness prior has been adopted and applied in nonlinear TAS. Nevertheless, there are also other useful priors such as sparseness and model-based prior which have not been investigated yet. This work aims to show how these priors can be implemented and included in the reconstruction process. Regularization through Bayesian formulation will be introduced specifically for this purpose, and a method for the determination of a proper regularization factor will be proposed. The comparative studies performed with different beam arrangements and regularization schemes on a few representative phantoms suggest that the beam optimization method developed for linear TAS also works for the nonlinear counterpart and the regularization scheme should be selected properly according to the available a priori information under specific application scenarios so as to achieve the best reconstruction fidelity. Though this work is conducted under the context of nonlinear TAS, it can also provide useful insights for other tomographic modalities. © 2017 Elsevier Ltd.

Fast quantitative MRI as a nonlinear tomography problem

Quantitative Magnetic Resonance Imaging (MRI) is based on a two-steps approach: estimation of the magnetic moments distribution inside the body, followed by a voxel-by-voxel quantification of the human tissue properties. This splitting simplifies the computations but poses several constraints on the measurement process, limiting its efficiency. Here, we perform quantitative MRI as a one step process; signal localization and parameter quantification are simultaneously obtained by the solution of a large scale nonlinear inversion problem based on first-principles. As a consequence, the constraints on the measurement process can be relaxed and acquisition schemes that are time efficient and widely available in clinical MRI scanners can be employed. We show that the nonlinear tomography approach is applicable to MRI and returns human tissue maps from very short experiments.

Hyperspectral tomography based on multi-mode absorption spectroscopy (MUMAS)

This paper demonstrates a hyperspectral tomographic technique that can recover the temperature and concentration field of gas flows based on multi-mode absorption spectroscopy (MUMAS). This method relies on the recently proposed concept of nonlinear tomography, which can take full advantage of the nonlinear dependency of MUMAS signals on temperature and enables 2D spatial resolution of MUMAS which is naturally a line-of-sight technique. The principles of MUMAS and nonlinear tomography, as well as the mathematical formulation of the inversion problem, are introduced. Proof-of-concept numerical demonstrations are presented using representative flame phantoms and assuming typical laser parameters. The results show that faithful reconstruction of temperature distribution is achievable when a signal-to-noise ratio of 20 is assumed. This method can potentially be extended to simultaneously reconstructing distributions of temperature and the concentration of multiple flame species.

3-D crustal velocity structure of western Turkey: Constraints from full-waveform tomography

The Sea of Marmara and western Turkey are characterized by intense seismicity and crustal deformation due to transition tectonics between the North Anatolian Fault Zone (NAFZ) and the extensional Aegean. Seismic imaging of the crust and uppermost mantle in W-NW Turkey is crucial to obtain a better understanding of its seismotectonics and geodynamics. So far, the Sea of Marmara and surroundings were considered in various active and passive seismic experiments providing significant information on crustal properties. Here, we further investigate the 3-D seismic velocity structure in this rapidly deforming region using non-linear full-waveform tomography based on the adjoint method. Our model is constrained by complete waveforms of 62 regional earthquakes (epicentral distance<10°) with magnitudes Mw≥3.7, which occurred between 2007 and 2015. Validation tests show that our final 3-D Earth model is able to explain seismic waveforms from earthquakes not used in the inversion at periods from 8-100s to within the data uncertainties. Furthermore, quantitative resolution analyses yield 15 to 35km horizontal resolution lengths in the shallow and deep crust beneath well-covered areas of W-NW Turkey. Our full-waveform tomography results indicate the presence of strong lateral and vertical velocity variations (2.55≤VS≤4.0km/s) down to depths of ∼35km. The seismic velocity distribution is characteristic of highly deformed and distributed crustal features along major fault zones (e.g. NAFZ and its branches), historic and recent regional volcanism (e.g. Kula volcanic province), and metamorphic core complex developments (e.g. Menderes and Kazdağ massifs). Radial anisotropy is very strong (around 20%) throughout the crust, further attesting to strong deformation and heterogeneity. Generally, our 3-D model is overall consistent with the active tectonics of western Turkey.

Benchmark evaluation of inversion algorithms for tomographic absorption spectroscopy.

Tomographic absorption spectroscopy (TAS) is experiencing a surge of interest due to recent progress in laser technology and advanced imaging concepts such as nonlinear tomography and compressed sensing. Nevertheless, even though numerous algorithms have been adapted from other tomographic areas such as medical imaging and engineering process control, and applied in TAS applications, systematic comparison between those methods has not been investigated. In this work, we aim to test major inversion algorithms on both the so-called rank-deficient (RD) and discrete ill-posed (DIP) problems. Comparative studies were performed on extensive cases, and the results for three representative phantoms are reported here. Other important topics such as determination of control parameters and the semiconvergent behavior of iterative methods have also been studied. According to our study, Landweber outperformed other methods for two RD cases and is slightly inferior to the maximum likelihood expectation maximization (MLEM) algorithm for the third one. It has also been found that Landweber and MLEM feature better immunity to semiconvergence than the others; however, since MLEM is sensitive to an initial guess, it is less robust than Landweber. On the other hand, the truncated singular value decomposition (TSVD) method is recommended for DIP problems due to its superior performance as it can effectively suppress detrimental effects from noise, which will be amplified during the inversion procedure. It should be emphasized that even though this study was conducted under the context of TAS, we expect it to provide useful insights for other tomographic modalities.

High power double-scale pulses from a gain-guided double-clad fiber laser

Generation of high power double-scale pulses from a gain-guided double-clad fiber laser is experimentally demonstrated. By employing the Yb-doped 10/130 double-clad fiber as the gain medium, the laser realizes an output power of 5.1 W and pulse energy of 0.175 µJ at repetition rate of 29.14 MHz. To the best of our knowledge, this average output power is the highest among the reported double-scale pulse oscillators. The autocorrelation trace of pulses contains the short (98 fs) and long (29.5 ps) components, and the spectral bandwidth of the pulse is 27.3 nm. Such double-scale pulses are well suited for seeding the high power MOPA (master oscillator power amplifier) systems, nonlinear frequency conversion and optical coherence tomography.

Exploiting surface consistency for surface-wave characterization and mitigation — Part 1: Theory and 2D examples

We have developed a new tomographic inversion method that is able to determine the properties of complex surface waves, which are multimodal and heterogeneous. These properties can be used to generate a detailed near-surface earth model or to predict and remove the surface waves, while protecting reflection signals even with aliased data. The inversion assumes plane-wave physics and generates surface-consistent model parameters as a function of frequency. In this paper, we validate our method with 2D models and data. In a companion paper, we demonstrate its application to 3D data. Inversion for a single mode is linear, but the linearity does not hold at higher frequencies, where multiple modes interfere. However, single-mode inversion results can be used to create a starting model for the subsequent nonlinear multimode tomography. The resulting velocity-frequency grid has greater resolution compared with a beam-forming method. The dispersion curves can be used as input to a subsequent standard 1D surface-wave inversion to generate a velocity-depth model. The tomographic method also determines a grid of attenuation quality factors and variations in the source amplitude and bandwidth, which correlate with the near-surface elevation changes. The amplitude and phase properties can be used together to predict the surface-wave waveforms, which can then be adaptively subtracted from the data on a trace-to-trace basis.

Transdimensional Love-wave tomography of the British Isles and shear-velocity structure of the East Irish Sea Basin from ambient-noise interferometry

We present the first Love-wave group velocity and shear velocity maps of the British Isles obtained from ambient noise interferometry and fully non-linear inversion. We computed interferometric inter-station Green's functions by cross-correlating the transverse component of ambient noise records retrieved by 61 seismic stations across the UK and Ireland. Group velocity measurements along each possible inter-station path were obtained using frequency-time analysis and converted into a series of inter-station traveltime datasets between 4 and 15 seconds period. Traveltime uncertainties estimated from the standard deviation of dispersion curves constructed by stacking randomly-selected subsets of daily cross-correlations, were observed to be too low to allow reasonable data fits to be obtained during tomography. Data uncertainties were therefore estimated again during the inversion as distance-dependent functionals. We produced Love-wave group velocity maps within 8 different period bands using a fully non-linear tomography method which combines the transdimensional reversible-jump Markov chain Monte Carlo (rj-McMC) algorithm with an eikonal raytracer. By modelling exact raypaths at each step of the Markov chain we ensured that the non-linear character of the inverse problem was fully and correctly accounted for. Between 4 and 10 seconds period, the group velocity maps show remarkable agreement with the known geology of the British Isles and correctly identify a number of low-velocity sedimentary basins and high-velocity features. Longer period maps, in which most sedimentary basins are not visible, are instead mainly representative of basement rocks. In a second stage of our study we used the results of tomography to produce a series of Love-wave group velocity dispersion curves across a grid of geographical points focussed around the East Irish Sea sedimentary basin. We then independently inverted each curve using a similar rj-McMC algorithm to obtain a series of one-dimensional shear velocity profiles. By merging all 1D profiles, we created a fully three-dimensional model of the crust beneath the East Irish Sea. The depth to basement in this model compares well with that averaged from seismic reflection profiles. This result is the first 3-dimensional model in the UK with fully quantified uncertainties: it shows basin depths and basement structures, and their concomitant uncertainties.

Robust total-variation based geophysical inversion using split Bregman and proximity operators

In this paper, we take advantages of split Bregman and proximity operators to formulate geophysical inverse problems in a convex optimization setting. In this type of formulation, we can efficiently consider a general convex data misfit term in order to incorporate more realistic error probability assumptions than the ordinary Gaussian distribution. Furthermore, we can simply impose convex non-quadratic and non-smooth regularization terms as a prior information.Although the proposed formulation can be extended for incorporating any types of convex regularization and data fidelity terms, here we consider misfit term corresponding to the Huber norm and anisotropic total variation regularization as a prior to an unknown model structure. This type of regularization enables us to construct piecewise-constant solutions.Two-dimensional linear and non-linear seismic travel-time tomography are studied to evaluate the efficiency of the proposed formulation for handling a robust measure of the misfit term in recovering blocky solutions. We consider the first arrival travel-times which are contaminated by outliers and also a mixed combination of Gaussian and outliers. Finally, Seinsfeld cross-hole tomography data set is used to investigate the performance of the robust approach on real data sets.

Development and validation of a reconstruction algorithm for three-dimensional nonlinear tomography problems.

This work reports the development and experimental validation of a reconstruction algorithm for three-dimensional (3D) nonlinear tomography problems. Many optical tomography problems encountered in practice are nonlinear, for example, due to significant absorption, multiple-scattering, or radiation trapping. Past research efforts have predominately focused on reconstruction algorithms for linear problems, and these algorithms are not readily extendable to nonlinear problems due to several challenges. These challenges include the computational cost caused by the nonlinearity (which was compounded by the large scale of the problems when they are 3D), the limited view angles available in many practical applications, and the measurement uncertainty. A new algorithm was therefore developed to overcome these challenges. The algorithm was validated both numerically and experimentally, and was demonstrated to be able to solve a range of nonlinear tomography problems with significantly enhanced efficiency and accuracy compared to existing algorithms.

Deterministic versus stochastic level-set regularization in nonlinear phase contrast tomography

A new nonlinear level-set regularization method to reconstruct the complex refractive index distribution with in-line phase contrast tomography measurements is presented under the assumption that the index is piecewise constant. The nonlinear iterative approach is based on the Fréchet derivative of the intensity recorded at a single propagation distance and for several projection angles. The algorithm is successfully applied to a multi-material object for several noise levels. Better reconstruction results are achieved with a stochastic perturbation of the level-set function. This evolution corresponds to a stochastic evolution of the shape of the reconstructed regions. The reconstruction errors can be further decreased with topological derivatives. The different algorithms are tested on various multi-material objects.

Role of nonlinear interactions in third-order acoustic tomography

In order to study the practical possibilities of acoustic nonlinear tomography, the paper describes wave generation processes caused by purely third-order nonlinear interaction and second-order double interaction. The occurring nonlinear secondary sources and nonlinear scattered fields are classified based on the character of their origin.